There are three products P, Q, and R, that may be shipped to customers using a single truck with load carrying capacity of 50K pounds (lbs) total weight. Shipment of each product has a $ value (e.g., profit) and a weight in lbs. Suppose the (weight, value) for products P, Q, and R is given by (10K lbs, $60K), (20K lbs, $100K), and (30K lbs, $120K), respectively. [You may assume that products are being shipped in predetermined quantities which have been factored into the (weight, value) numbers provided above. You cannot ship more of a product than specified above. Further, unless stated otherwise, partial shipments of fractional amounts of a product are not allowed, i.e., either a product is shipped in its entirety or not at all.]

1. (*10 points) Bottleneck Ratio Method: Define a bottleneck ratio for this PQ problem. Use your bottleneck ratio values to prioritize products and decide which products to ship on the 50K-lbs truck. What is the total value of your shipment on the truck? How does the solution change if partial shipments of a product are allowed with proportional reduction in value?

2. (*5 points) Optimality Maximizing Shipped Value: Does any solution to part 1 from your bottleneck ratio approach maximize shipped value? If yes or if unsure, explain. If not, specify a better solution (for the items to ship) and its shipment value.

3. (*10 points) Weight, Volume: Suppose in addition to weight, each product has a volume of space that it occupies. Suppose the truck can carry 1500 cubic feet of materials and products P, Q, and R occupy, respectively, 600, 2500, 500 cu. ft. of truck space. Show how the bottleneck ratio method can incorporate this new information by presenting your new solution of which products to ship. Clearly mention the bottleneck resource(s) for your solution and its shipment value.

(5 points) PQ Lesson: What do you think is the single most important lesson from the PQ problem? Why did you pick this lesson as most important?

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